# Analysis and Synthesis in the Design of Magnetic Switching Electric Machines

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, …, X

_{k}are called factors, the analysis of which is carried out in the k-dimensional factor space. The initial information for constructing the response functions is formed as a result of various combinations of variable values (factors), each of which occupies one of three levels: lower, upper or middle (base). The coefficients included in the response function are called regression coefficients. To simplify the analysis, some of the terms are excluded from the regression equation; however, the remaining terms must provide the necessary accuracy of the approximation. The polynomials obtained in this way are used in optimization algorithms.

## 2. Materials and Methods

_{am}, kW/kg. In other words, it is the active power per unit mass of only that structural part of the machine that contains magnetic and electrical circuits.

## 3. Results

#### 3.1. Features of the Design Model of an Electric Machine

_{v}—the cross-sectional area of copper of the excitation winding:

_{pc}= 1.5 ÷ 2.0 is the scattering coefficient of the magnetic flux; κ

_{zp}= 1.15 is the safety factor for the regulation of the generator’s voltage; k

_{F}= 1.25 ÷ 1.45 is the ratio of the magneto motive force of the magnetic circuit to that of the gap; B

_{δ}= B

_{z}

_{1}κ

_{zc}α

_{z}is the gap induction; B

_{z}

_{1}is the admissible value of induction in the teeth of the inductor; κ

_{zc}= 0.81 ÷ 0.99 is the coefficient of filling with steel; δ’ = 4(I − 1) δ is the calculated value of the air gap (i is the number of the axial layers or disks of the stator); µ

_{0}= 4π·10

^{−7}H/m is the magnetic constant; k

_{zm}= 0.35 ÷ 0.65 is the coefficient of filling the window with copper; j—the current density in windings.

_{n}cosφ

_{n}is the calculated power given during the design; E = (1.15 … 1.3)U

_{n}is the EMF for the generator, and E = (0.7 … 0.85)U

_{n}is the EMF for the engine; U

_{n}is the rated voltage; I

_{n}is the rated current; cosφ

_{n}is the rated power factor; k

_{λ}= λ

_{max}/λ

_{min}= 1.5 ÷ 2 is the magnetic flux modulation index; b

_{z}= b

_{z min}= (3 ÷ 5)·10

^{−3}is the minimum permissible tooth width; f is the frequency of electric current; n is the frequency of the electric current; rotor speed.

^{2}− (S

_{v}/3)

^{3}≥ 0, one should use the Cardano solution [31], in which only the first real root has a physical meaning:

_{z}

_{1}and h

_{z}

_{2}are the axial dimensions of the teeth of the inductor and the rotor, respectively.

_{z}are set from the condition of the optimal ratio of copper and steel.

_{z}= 0.32a.

_{i}, the width of the teeth of the inductor b

_{z}

_{1}and the teeth of the rotor are taken b

_{z}

_{2}, calculated from the average diameter of the inductor D

_{cp}= 0.5 (D

_{v}+ D

_{n}).

_{m}and g

_{C}are the specific indicators of the mass of copper and steel, respectively.

_{δ}and the modulation coefficient of the main flux k

_{λ}.

_{δ}and k

_{λ}are refined. In the next block, the geometric dimensions of the machine module are finally determined.

_{δ}and its modulation k

_{λ}.

#### 3.2. Analysis of Experiments and Significant Factors

_{n}; rated voltage U

_{n}; rated current frequency f; rated speed n

_{n}; induction in the air gap B

_{δ}; saturation induction in the stator tooth B

_{nas}; excitation winding voltage U

_{v}. The rest of the design parameters were assigned fixed values, determined from the experience of designing electrical machines and the parameters of the selected electrical materials.

_{n}; induction in the air gap B

_{δ}; saturation induction in the stator tooth B

_{nas}.

_{i}is the coded value of the factor; ${\tilde{X}}_{i}$—the natural value of the factor; ${\tilde{X}}_{io}$—the natural value of the main level of the factor; I

_{i}—the factor variation interval; i—the factor number.

_{nas}by X

_{1}, the factor f by X

_{2}, the factor n

_{n}by X

_{3}, the factor B

_{δ}by X

_{4}.

#### 3.3. Complete Factorial Experiment and Calculation Model Optimization

^{k}. Here N is the number of numerical experiments; k is the number of factors (in our case, 4); 2—the number of levels.

_{1}, X

_{2}, …, X

_{k}). Selecting a model means choosing the type of this function, writing down its equation.

## 4. Discussion

_{am}, kW/kg from the average value (Table 2), the reproducibility variance is calculated according to the data of parallel observations of the planning matrix plan at each point. After that, the homogeneity of dispersions is checked according to the Cochran test. The significance of the regression coefficients in the mathematical model is determined by the Student’s criterion. Further, according to Fisher’s criterion, the adequacy of the model is checked. If the model is adequate, then you can proceed to a steep ascent.

_{n}= 750 V; rated current I

_{n}= 148 A; power factor cosφ

_{n}= 0.9.

## 5. Conclusions

## 6. Patents

- Afonin A., German-Galkin S., Cierzniecki P., Hrynkiewicz J., Kramarz W., Szymczak P. Modular reluctance machine. PCT Int. Public Nomber WOo 1/03270.Al. Int. Public. Date 11 January 2001. Priority data 22 May 1999.
- Шaйтop M.M., Aφoнiн A.O., Pяcкoв Ю.I. Eлeктpoмexaнiчнa cиcтeмa aкciaльнo-paдiaльнoi кoнφiгypaцii. Пaтeнт UA 49630. Oпyбликoвaнo: 15 December 2003. Бaзa пaтeнтiв Укpaїни. Available online: URL uapatents.com (accessed on 25 May 2021).
- Шaйтop M.M., Pяcкoв Ю.I., Бepeзoвeнкo O.B. Maгнiтoкoмyтyючий двoмepeжeвий гeнepaтop з caмoзбyджeнням. Пaтeнт UA 62802. Oпyбликoвaнo: 15 December 2003. Бaзa пaтeнтiв Укpaїни. Available online: URL uapatents.com (accessed on 25 May 2021).
- Шaйтop M.M., Pяcкoв Ю.I., Шeвцoв E.I. Tpиφaзний двигyн aкciaльнo-paдiaльнoi кoнφiгypaцii. Пaтeнт UA 69069. Oпyбликoвaнo: 16 August 2004. Бaзa пaтeнтiв Укpaїни. Available online: URL uapatents.com (accessed on 25 May 2021).
- Шaйтop M.M., Pяcкoв Ю.I., Жидкoв B.O. Бeзшaтyнoвий бeзщiткoвий гeнepaтopний aгpeгaт. Пaтeнт UA 68262. Oпyбликoвaнo: 15.07.2004. Бaзa пaтeнтiв Укpaїни. Available online: URL uapatents.com (accessed on 25 May 2021).
- Шaйтop M.M., Pяcкoв Ю.I. Maгнiтoкoмyтyючa диcкoвa мaшинa з кiгтeпoдiбним iндyктopoм. Пaтeнт UA 62556. Oпyбликoвaнo: 15 December 2003. Бaзa пaтeнтiв Укpaїни. Available online: URL uapatents.com (accessed on 25 May 2021).
- German-Galkin S., Hrynkiewicz J. Uklad modulu elektromechanicznego maszyny elektrycznej. Pat. UP RP z 20 December 2013r. Nr P.389456. Zglosz, pat. 03 November 2009.
- Бopмoтaв A.B., Гepмaн-Гaлкин C.Г., Зaгaшвили Ю.B., Лeбeдeв B.B. Moдyльнaя элeктpичecкaя мaшинa. Пaтeнт RU 2,510,121 C2, 20 March 2014.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The elementary module of a machine with magnetic commutation: (

**a**) principle of operation; (

**b**) designations of geometric dimensions.

**Figure 6.**Dependences of the specific power p

_{am}on the current frequency at various generator rotation values: n

_{1}= 10 r/s; n

_{2}= 25 r/s; n

_{3}= 50 r/s; n

_{4}= 100 r/s; n

_{5}= 300 r/s; n

_{6}= 400 r/s.

**Figure 7.**Dependences of the specific power p

_{am}on rotation frequency at different frequencies of current: f

_{1}= 400 Hz; f

_{2}= 1000 Hz; f

_{3}= 2000 Hz; f

_{4}= 3000 Hz; f

_{5}= 5000 Hz; f

_{6}= 10,000 Hz.

**Figure 8.**Dependence of the specific power p

_{am}on the value of induction in the air gap at a constant generator frequency: f

_{1}= 400 Hz; f

_{2}= 1000 Hz; f

_{3}= 2000 Hz; f

_{4}= 3000 Hz; f

_{5}= 4000 Hz; f

_{6}= 5000 Hz; f

_{7}= 10,000 Hz.

**Figure 9.**Dependences of the specific power p

_{am}on the induction of the saturation of the tooth and induction in the air gap.

Factors | Levels | Variation Interval I | Units | ||
---|---|---|---|---|---|

−1 | 0 | +1 | |||

X_{1} | 1.6 | 1.9 | 2.2 | 0.3 | T |

X_{2} | 3000 | 5000 | 7000 | 2000 | Hz |

X_{3} | 310 | 350 | 390 | 40 | rev/s |

X_{4} | 0.3 | 0.4 | 0.5 | 0.1 | T |

№ | X_{0} | X_{1} | X_{2} | X_{3} | X_{4} | X_{1×2} | X_{1×3} | X_{1×4} | X_{2×3} | X_{2×4} | X_{3×4} | p_{am}, kW/kg | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Code | Code | Code | Code | Code | Code | Code | Code | Code | Code | Code | Y_{ν,1} | Y_{ν,2} | |

1 | + | - | - | - | - | + | + | + | + | + | + | 1.267 | 1.365 |

2 | + | + | - | - | - | - | - | - | + | + | + | 1.113 | 1.164 |

3 | + | - | + | - | - | - | + | + | - | - | + | 1.628 | 1.607 |

4 | + | + | + | - | - | + | - | - | - | - | + | 1.410 | 1.464 |

5 | + | - | - | + | - | + | - | + | - | + | - | 1.327 | 1.270 |

6 | + | + | - | + | - | - | + | - | - | + | - | 1.079 | 1.068 |

7 | + | - | + | + | - | - | - | + | + | - | - | 1.812 | 1.789 |

8 | + | + | + | + | - | + | + | - | + | - | - | 1.547 | 1.530 |

9 | + | - | - | - | + | + | + | - | + | - | - | 0.936 | 0.948 |

10 | + | + | - | - | + | - | - | + | + | - | - | 1.461 | 1.178 |

11 | + | - | + | - | + | - | + | - | - | + | - | 1.346 | 1.421 |

12 | + | + | + | - | + | + | - | + | - | + | - | 1.116 | 1.115 |

13 | + | - | - | + | + | + | - | - | - | - | + | 0.906 | 0.953 |

14 | + | + | - | + | + | - | + | + | - | - | + | 1.180 | 1.141 |

15 | + | - | + | + | + | - | - | - | + | + | + | 0.923 | 0.919 |

16 | + | + | + | + | + | + | + | + | + | + | + | 1.144 | 1.186 |

Inductor Generator | Current Frequency Hz | Rotation Frequency rpm | Diameter m | Length m | Specific Power kW/kg | Efficiency |
---|---|---|---|---|---|---|

Classic Drum Rotor Design | 2700 | 3000 | 0.740 | 0.432 | 0.10 | 0.89 |

Axial-Radial Configuration (Project Without Parametric Synthesis) | 2700 | 3000 | 0.836 | 0.366 | 0.43 | 0.88 |

Axial-Radial Configuration (Parametric Synthesis) | 5700 | 22,200 | 0.338 | 0.355 | 1.89 | 0.93 |

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**MDPI and ACS Style**

Shaitor, N.; Kelemen, M.; Yakimovich, B.
Analysis and Synthesis in the Design of Magnetic Switching Electric Machines. *Actuators* **2021**, *10*, 164.
https://doi.org/10.3390/act10070164

**AMA Style**

Shaitor N, Kelemen M, Yakimovich B.
Analysis and Synthesis in the Design of Magnetic Switching Electric Machines. *Actuators*. 2021; 10(7):164.
https://doi.org/10.3390/act10070164

**Chicago/Turabian Style**

Shaitor, Nikolay, Michal Kelemen, and Boris Yakimovich.
2021. "Analysis and Synthesis in the Design of Magnetic Switching Electric Machines" *Actuators* 10, no. 7: 164.
https://doi.org/10.3390/act10070164